What if you had a quadratic expression like
or
in which some or all the coefficients were negative? How could you factor that expression? After completing this Concept, you'll be able to factor quadratic expressions like these for various negative coefficient values.

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Guidance

In the previous concept, we saw how to factor quadratic expressions whose coefficients were all positive. In this concept we will now see what happens when we factor quadratic expressions where some of the coefficients are negative.

Factor when a = 1, b is Negative and c is Positive

Now let’s see how this method works if the middle coefficient is negative.

Example A

Factor
.

Solution

We are looking for an answer that is a product of two binomials in parentheses:

When negative coefficients are involved, we have to remember that negative factors may be involved also. The number 8 can be written as the product of the following numbers:

but also

and

but also

The last option is the correct choice. The answer is
. We can check to see if this is correct by multiplying
:

The answer checks out.

Example B

Factor
.

Solution

We are looking for an answer that is a product of two binomials in parentheses:

The number 16 can be written as the product of the following numbers:

The answer is
.

In general, whenever
is negative and
and
are positive, the two binomial factors will have minus signs instead of plus signs.

Factor when a = 1 and c is Negative

Now let’s see how this method works if the constant term is negative.

Example C

Factor
.

Solution

We are looking for an answer that is a product of two binomials in parentheses:

Once again, we must take the negative sign into account. The number -15 can be written as the product of the following numbers:

The answer is
.

We can check to see if this is correct by multiplying:

The answer checks out.

Example D

Factor
.

Solution

We are looking for an answer that is a product of two binomials in parentheses:

The number -24 can be written as the product of the following numbers:

The answer is
.

Factor when a = - 1

When
, the best strategy is to factor the common factor of -1 from all the terms in the quadratic polynomial and then apply the methods you learned so far in this section

Example E

Factor
.

Solution

First factor the common factor of -1 from each term in the trinomial. Factoring -1 just changes the signs of each term in the expression: